Algorithm 1 Diameter-regularized Hessian weighted K-means in vector case

Input: Weights vector $\{{\mathbf{w}}^{\left(1\right)},\cdots ,{\mathbf{w}}^{\left(d^{\prime }\right)}\}$, Hessian matrices $\{H^{\left(1\right)},\cdots ,H^{\left(d^{\prime }\right)}\}$, diameter regularizer $\beta >0$, number of clusters K, iterations T  Initialize the K cluster centers $\{{\mathbf{c}}_0^{\left(1\right)},\cdots ,{\mathbf{c}}_0^{\left(K\right)}\}$ randomly  for $t=1$ to T do   Assignment step:   Initialize $C_t^{\left(k\right)}=\varnothing$ for all $k\in \left[K\right]$.   for $j=1$ to $d^{\prime }$ do    Assign ${\mathbf{w}}^{\left(j\right)}$ to the nearest cluster center, i.e., find $k_t^{\left(j\right)}=arg{min}_{k\in \left[K\right]}{\parallel {\mathbf{w}}^{\left(j\right)}-{\mathbf{c}}_{t-1}^{\left(k\right)}\parallel }_2^2$ and let $$\begin{array}{c}\hfill C_t^{\left(k_t^{\left(j\right)}\right)}\leftarrow C_t^{\left(k_t^{\left(j\right)}\right)}\cup \left\{{\mathbf{w}}^{\left(j\right)}\right\}\end{array}$$ (36)   end for   Update step:   Find current farthest pair of centers $(k_1,k_2)=arg{max}_{k_1,k_2}{\parallel {\mathbf{c}}_{t-1}^{\left(k_1\right)}-{\mathbf{c}}_{t-1}^{\left(k_2\right)}\parallel }_2^2$.   Update ${\mathbf{c}}_t^{\left(k_1\right)}$ and ${\mathbf{c}}_t^{\left(k_2\right)}$ by $$\begin{array}{c}\hfill {\mathbf{c}}_t^{\left(k_1\right)}={\left(\sum _{{\mathbf{w}}^{\left(j\right)}\in C_t^{\left(k_1\right)}}{H}^{\left(j\right)}+\beta I_m\right)}^{-1}\left(\sum _{{\mathbf{w}}^{\left(j\right)}\in C_t^{\left(k_1\right)}}{H}^{\left(j\right)}{\mathbf{w}}^{\left(j\right)}+\beta {\mathbf{c}}_t^{\left(k_2\right)}\right)\\ {\mathbf{c}}_t^{\left(k_2\right)}={\left(\sum _{{\mathbf{w}}^{\left(j\right)}\in C_t^{\left(k_2\right)}}{H}^{\left(j\right)}+\beta I_m\right)}^{-1}\left(\sum _{{\mathbf{w}}^{\left(j\right)}\in C_t^{\left(k_2\right)}}{H}^{\left(j\right)}{\mathbf{w}}^{\left(j\right)}+\beta {\mathbf{c}}_t^{\left(k_1\right)}\right)\end{array}$$ (37)   for $k=1$ to K, $k\notin \{k_1,k_2\}$ do   Update the cluster centers by $$\begin{array}{c}\hfill {\mathbf{c}}_t^{\left(k\right)}={\left(\sum _{{\mathbf{w}}^{\left(j\right)}\in C_t^{\left(k\right)}}{H}^{\left(j\right)}\right)}^{-1}\left(\sum _{{\mathbf{w}}^{\left(j\right)}\in C_t^{\left(k\right)}}{H}^{\left(j\right)}{\mathbf{w}}^{\left(j\right)}\right)\end{array}$$ (38)   end for  end for Output: centers $\{{\mathbf{c}}_T^{\left(1\right)},\cdots ,{\mathbf{c}}_T^{\left(K\right)}\}$ and assignments $\{C_T^{\left(1\right)},\cdots ,C_T^{\left(K\right)}\}$.